Spectral moment embedding: a matrix-moment generalisation of characteristic response analysis for covariance-based signal classification

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Covariance descriptors summarise cross-channel dependence in biosignal and machinery classification, but compact eigenvector summaries can map distinct covariance matrices to identical features. We define Spectral Moment Embedding (SME), a sign-invariant matrix-moment representation that preserves covariance information targeted by Characteristic Response Analysis (CRA) and its Positive-Orthant Characteristic Response (POCR), while avoiding non-injectivity from positive-orthant eigenvector summation. SME regularises each epoch covariance, forms C,C²,… ,Cᵖ by dense multiplication, and applies diagonal, upper-triangular half-vectorisation, or row-sum projections; a sub-window trajectory variant tests non-stationary trials. Five public datasets were evaluated under strict subject- or recording-grouped cross-validation against CRA/POCR, covariance, Riemannian, polynomial, and spectral baselines. SME exceeded POCR on Sleep-EDF staging ( $$64.5%$$ vs. $$44.2%$$ ), MIT-BIH arrhythmia ( $$37.7%$$ vs. $$19.2%$$ ), CWRU bearing faults ( $$84.1%$$ vs. $$45.5%$$ ), and UCI HAR ( $$80.5%$$ vs. $$62.0%$$ ). Motor imagery was negative: static SME reached $$30.2%$$ against POCR at $$31.9%$$ , and trajectory-SME fell to $$25.6%$$ . In sleep staging, multitaper PSD led globally ( $$71.9%$$ ), whereas SME retained a narrow N3 F1 advantage (0.806 vs. 0.782). Timing followed O(N³p) scaling and was three-to-five times faster than POCR at $$N=128$$ . SME is therefore a general covariance-moment algorithm, strongest when stable cross-channel covariance carries class information.